# Variation of parameters - i have different particular soluti

• omar yahia
In summary, the conversation is about solving a 3rd order ODE using the variation of parameters method. The solution involves finding the homogeneous solution and particular solution, where the particular solution is a combination of three integrals. The arrangement of the three functions used in the particular solution can affect the resulting answer, but it is possible for them to be equivalent. The speaker also mentions discovering a mistake in their solution, apologizing for trusting a senior without thorough revision.
omar yahia
i was trying to get a particular solution of a 3rd order ODE using the variation of parameters method
the homogeneous solution is yh = c1 e-x + c2 ex + c3 e2x
the particular solution is yp=y1u1+y2u2+y3u3
as u1=∫ (w1 g(x) /w) dx , u2=∫ (w2 g(x) /w) dx , u3=∫ (w3 g(x) /w) dx
w =
|y1 y2 y3|
|y'1 y'2 y'3|
|y''1 y''2 y''3|

when i choose y1 , y2 , y3 to be e-x,ex,e2x i get an answer ,
but when i change the arrangement (like: ex,e-x,e2x )
i get another different answer !

so , i have two questions
1 is it normal to have different results when changing who is y1 , y2 , y3 , or am i doing something wrong?
2 if it is normal , does that mean i can have too many different particular solutions , just by changing who is y1 , y2 , y3?

What are the different answers you get?
Maybe they are equivalent, and just look differently?

mfb said:
What are the different answers you get?
Maybe they are equivalent, and just look differently?

ummmmmm ... ahhhh .. actually ... when you asked me to show the results i went to prepare them for uploading
and as i was rewriting the solution i discovered an extra (2) multiplied in one little tiny term , will i fixed it and went on , and the results were the same indeed , i am terribly sorry for this mistake it happened because i trusted the solution of a senior without a thorough revision
thank you for your reply and i apologies again.

Never trust a senior!

## What is the variation of parameters method?

The variation of parameters method is a technique used to find a particular solution to a non-homogeneous linear differential equation. It involves using the known solutions of the associated homogeneous equation to construct a particular solution.

## How is the variation of parameters method different from the method of undetermined coefficients?

The method of undetermined coefficients is used to find a particular solution for specific types of non-homogeneous equations, while the variation of parameters method can be applied to a wider range of non-homogeneous equations.

## What are the steps of the variation of parameters method?

The steps of the variation of parameters method are: 1) finding the general solution to the associated homogeneous equation, 2) finding the Wronskian of the homogeneous solution, 3) finding the integrals of the non-homogeneous terms, and 4) using these integrals to construct the particular solution.

## When is the variation of parameters method most useful?

The variation of parameters method is most useful when the non-homogeneous terms in the differential equation are difficult to guess or cannot be represented by polynomials or exponential functions, which are typically required for the method of undetermined coefficients.

## What are some limitations of the variation of parameters method?

One limitation of the variation of parameters method is that it can be more time-consuming and complex compared to the method of undetermined coefficients. Additionally, the method may not always yield a closed-form solution, which can make it difficult to find the particular solution.

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